UNSTEADY MHD THREE DIMENSIONAL FLOW OF MAXWELL FLUID THROUGH POROUS MEDIUM IN A PARALLEL PLATE CHANNEL UNDER THE INFLUENCE OF INCLINED MAGNETIC FIELD

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1 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 UNSTEADY MHD THREE DIMENSIONAL FLOW OF MAXWELL FLUID THROUGH OROUS MEDIUM IN A ARALLEL LATE CHANNEL UNDER THE INFLUENCE OF INCLINED MAGNETIC FIELD L.Sreekala, M.VeeraKrishna *, L.HariKrishna and E.KesavaReddy Assistant rofessor, Department of Mathematics, CRIT, Anantapur, Andhra radesh, India Department of Mathematics, Rayalaseema University, Kurnool, Andhra radesh, India Assistant rofessor, Department of Mathematics, AITS, Kadapa, Andhra radesh, India rofessor, Department of Mathematics, JNTU, Anantapur, Andhra radesh, India ABSTRACT In this paper, we discuss the unsteady hydro magnetic flow of an electrically conducting Maxwell fluid in a parallel plate channel bounded by porous medium under the influence of a uniform magnetic field of strength Ho inclined at an angle of inclination with the normal to the boundaries. The perturbations are created by a constant pressure gradient along the plates. The time required for the transient state to decay and the ultimate steady state solution are discussed in detail. The exact solutions for the velocity of the Maxwell fluid consists of steady state are analytically derived, its behaviour computationally discussed with reference to the various governing parameters with the help of graphs. The shear stresses on the boundaries are also obtained analytically and their behaviour is computationally discussed in detail. KEYWORDS: Maxwell fluids, unsteady flows, porous medium, parallel plate channels, MHD flows I. INTRODUCTION Several fluids including butter, cosmetics and toiletries, paints, lubricants, certain oils, blood, mud, jams, jellies, shampoo, soaps, soups, and marmalades have rheological characteristics and are referred to as the non-newtonian fluids. The rheological properties of all these fluids cannot be explained by using a single constitutive relationship between stress and shear rate which is quite different than the viscous fluids [, ]. Such understanding of the non-newtonian fluids forced researchers to propose more models of non-newtonian fluids. In general, the classification of the non-newtonian fluid models is given under three categories which are called the differential, the rate, and the integral types []. Out of these, the differential and rate types have been studied in more detail. In the present analysis we discuss the Maxwell fluid which is the subclass of rate-type fluids which take the relaxation phenomenon into consideration. It was employed to study various problems due to its relatively simple structure. Moreover, one can reasonably hope to obtain exact solutions from Maxwell fluid. This motivates us to choose the Maxwell model in this study. The exact solutions are important as these provide standard reference for checking the accuracy of many approximate solutions which can be numerical or empirical in nature. They can also be used as tests for verifying numerical schemes that are being developed for studying more complex flow problems [ 9]. On the other hand, these equations in the non-newtonian fluids offer exciting challenges to mathematical physicists for their exact solutions. The equations become more problematic, when a non-newtonian fluid is discussed in the presence of MHD and porous medium. Despite this fact, various researchers are still making their interesting contributions in the field (e.g., see some recent studies [ ]. Few investigations which provide the examination of non-newtonian fluids in a rotating frame are also Vol., Issue, pp. -

2 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 presented [ 9]. Recently Faisal Salah [] discussed two explicit examples of acceleration subject to a rigid plate are taken into account. Constitutive equations of a Maxwell fluid are used and modified Darcy s law has been utilied. The exact solutions to the resulting problem are developed by Fourier sine transform. With respect to physical applications, the graphs are plotted in order to illustrate the variations of embedded flow parameters. The mathematical results of many existing situations are shown as the special cases of that study. Such studies have special relevance in meteorology, geophysics, and astrophysics. Hayat et.al [] investigated to analye the MHD rotating flow of a Maxwell fluid through a porous medium in parallel plate channel. M.V. Krishna [] discussed analytical solution for the unsteady MHD flow is constructed in a rotating non-newtonian fluid through a porous medium taking hall current into account. In this paper, we examine the MHD flow of Maxwell fluid through a porous medium in a parallel plate channel with inclined magnetic field, the perturbations in the flow are created by a constant pressure gradient along the plates. The time required for the transient effects to decay and the ultimate steady state solution are discussed in detail. The exact solutions of the velocity in the Maxwell fluid consists of steady state are analytically derived, its behaviour computationally discussed with reference to the various governing parameters with the help of graphs. The shear stresses on the boundaries are also obtained analytically and their behaviour is computationally discussed. II. FORMULATION AND SOLUTION OF THE ROBLEM We consider the unsteady flow of an electrically conducting Maxwell fluid through porous medium in a parallel plate channel subjected to a uniform transverse magnetic field of strength H o inclined at an angle of inclination normal to the channel walls. The boundary plates are assumed to be parallel to xy-plane and the magnetic field to the -axis in the transverse x-plane. The component along - direction induces a secondary flow in that direction while its x-components changes perturbation to the axial flow. At the fluid is driven by a prescribed pressure gradient parallel to the channel walls. We choose a Cartesian system O(x, y, such that the boundary walls are at and, since the plates extends to infinity along x and y directions, all the physical quantities except the pressure depend on and t alone. The unsteady hydro magnetic equations governing the electrically conducting Maxwell fluid under the influence of transverse magnetic field with reference to a frame are V ρ ( V. V p div S J B R t (. V. (.. B (. B J m (. t B E (. t Where, J is the current density, B is the total magnetic field, E is the total electric field, m is the magnetic permeability, V = (u, v, w is the velocity field, T is the Cauchy stress tensor, B is the total magnetic field so that B=B Sin + b, where B is the applied magnetic field parallel to the -axis and b is the induced magnetic field. The induced magnetic field is negligible so that the total magnetic field B = (,, B Sin, the Lorent force J B B Sin V, is the electrical conductivity D of the fluid, ρ is the density of the fluid, and is the material derivative and R is the Darcy Dt resistance. The extra tensor S for a Maxwell fluid is T p I S (. DS T S λ LS SL μa (. Dt l 8 Vol., Issue, pp. -

3 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 p I where is the stress due to constraint of the impermeability, here p is the static fluid pressure, I is the identity tensor, is the viscosity of the fluid, is the material time constants referred to as relaxation time, it is assumed that. The first Rivlin-Ericksen tensor A is defined as A = (grad V + (grad V T (.8 It should be noted that this model includes the viscous Navier-Stokes fluid as a special case for. Let us indicate the stress tensor and the velocity component as V(, t = (u,, w (.9 According to Tan and Masuoka [] Darcy s resistance in an Oldroyd-B fluid satisfies the following expression: R r V t k t (. where λ r is the retardation time, is the porosity (< <, and k is the permeability of the porous medium. For Maxwell fluid λ r=, and hence, R V t k (. Making use of the equations (., (. and (.8, the equation (. reduces to u p S x ρ σb Sin u R x t x (. w S y ρ σb Sin w R t (. Where R x and R are x and -components of Darcy s resistance R; u w S and x S y t t (. The equations (. and (. reduces to u p S μφ x ρ σb Sin u u t x k (. w S y μφ ρ σb Sin v w t k (. Let quiw Combining equations (. and (., we obtain q p μφ ρ ( S is σb Sin q q x y t x k (. Since q ( S is x y t (.8 Substituting the equation (.8 in the equation (., we obtain the equation for the governing the flow through a porous medium with respect to the rotating frame is given by q σb Sin ν p q λ ( λ q λ ν t t ρ k t ρ t x (.9 The boundary and initial conditions are q t, (. q, t, l (. q(,t, dq(,t, dt t, for all (. We introduce the following non dimensional variables are * * q l * tν * ωl * ξ * l, q, t,ω, ξ, l ν l ν l ρν μ 9 Vol., Issue, pp. - λ

4 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 Using non dimensional variables the governing equations are (dropping asterisks in all forms q q β M Sin D β q β ( (. t t t ρ t σμe H l where, M l is the Hartmann number, D ρv k is the inverse Darcy arameter, λν p β is the material parameter related to relaxation time and is the pressure gradient. l x The corresponding initial and boundary conditions are q t, (. q, t, (. q(,t, dq(,t, dt t, for all (. supposing the pressure is given by i t e, t,, t (. Taking Laplace transforms of equations (. and (. using initial conditions (. the governing equations in terms of the transformed variable reduces to d q s ( ( M Sin D s ( M Sin D q d ( i ω s - iω s (.8 Solving equation (.8 subjected to the conditions (. and (., we obtain ( iβ ω Cosh( λ Cosh(λ q λ (s iω λ s ( iβ ω Cosh( λ Sinh(λ Cosh( λ.sinh(λ λ (s iω.sinh(λ λ s.sinh(λ ( iβ ω.sinh(λ.sinh(λ ( iβ ω λ (s iω. Sinh(λ λ s.sinh(λ λ (s iω λ s( s (.9 Where λ βs ( β (M Sin D φs (M Sin D φ Taking the inverse Laplace transforms to the equations (.9 on both sides, We obtain Cosh(b Cosh(b.Sinh(b Sinh(b q b b Sinh(b b Sinh(b b Cosh(b.Sinh(b ( iβω Cosh(b (iω s (iω s Sinh(b Sinh(b i ω ( iβ ω.cosh(b t e Sinh(b (s s (s iω ( iβω.cosh(b.sinh(b ( iβω.sinh(b (s s (s iω.sinh(b (s s (s iω Sinh(b ( iβ ω Cosh(b Cosh(b.Sinh(b (s s (s iω (s s s (s s (s.sinh(b (s.sinh(b s (s s t (s.sinh(b s e (s Vol., Issue, pp. -

5 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 ( iβω.cosh(b (s s (s iω ( iβω.cosh(b.sinh(b ( iβω.sinh(b (s s (s iω.sinh(b (s s (s iω Sinh(b ( iβ ω Cosh(b Cosh(b.Sinh(b (s s (s iω (s s s (s s s.sinh(b (s n n.sinh(b s t e s s.sinh(b (s s s ( iβ ω.cosh(b.sinh(b ( iβ ω. Sinh(b b (s s (s iω b (s s (s iω s t b (s s (s b (s s (s Cosh(b.Sinh(b Sinh(b e ( iβ ω.cosh(b.sinh(b ( iβ ω. Sinh(b b (s s (s iω b (s s (s iω Cosh(b.Sinh(b Sinh(b s t e (. b (s s (s b (s s (s (Where the constants are mentioned in the appendix The shear stresses on the upper and lower plate are given by dq dq τu and τ L (. d d III. RESULTS AND DISCUSSION We discuss the unsteady flow of an electrically conducting Maxwell fluid through a porous medium in parallel plate channel subjected to uniform magnetic field. In unperturbed state the perturbation are created by performing to imposition of constant pressure gradient along the axis (OX of the channel walls the velocity component along the imposed pressure gradient and normal to it. Under the boundary layer assumptions these velocity components are to functions of and t alone, where corresponds to the direction of axis of the channel. The transverse magnetic field once arising give rise to Lorent forces resisting the flow along normal to the channel wall. The constitutive equations relating the stress and rate of strain are chosen to depict the Maxwell fluid. The Brinkman s model has been chosen to analyses the flow through a porous medium. The equation governing the velocity components and with reference to frame ultimately can be combined into a single equation by defining the complex velocity q u iw The expression for the components of the stresses are manipulated from the stress and strain relationships. Under these assumptions the ultimate governing equations for the unsteady flow through a porous medium with reference to frame is formulated the corresponding boundary and initial conditions. This boundary value problem has been solved using non-dimensional variables making use of Laplace transform technique. The solution for the combined velocity q consists of two kinds of terms. Steady state. The transient terms involving exponentially varying time dependence. The analysis of transient terms indicates that this transient velocity decay exponentially in dimensionless time to of order i.e., t max,, s s. This decay in the transient term depends on the non-dimensional parameters β, M and D -. When these transient terms decay the ultimate velocity consists of steady and oscillatory components. Vol., Issue, pp. -

6 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 Cosh(b Cosh(b.Sinh(b Sinh(b ( q (. steady b b Sinh(b b Sinh(b b The flow governed by the non-dimensional parameters namely vi. M the magnetic field parameter (the Hartmann number, D - the inverse Darcy parameter, is the material time parameter referred as relaxation time. The computational analysis has been carried out to discuss the behaviour of velocity components u and w on the flow in the rotating parallel plate channel and the lower plate executes non-torsional oscillations in its own plane with reference to variations in the governing parameters may be analyed from figures (- and (- respectively ( = =, t =., /, /. We may note that the effect of the magnetic field on the flow from figures ( and. The magnitude of the velocity component u reduces and the velocity component w increases with increase in the Hartmann number M. However, the resultant velocity reduces throughout the fluid region with increase in the intensity of the magnetic field (the Hartmann number M. The figures ( and represent the velocity profiles with different variation in the inverse Darcy parameter D -. We find that the magnitude of u reduces with decrease in the permeability of the porous medium, while the magnitude of w experiences a slight enhancement with increase in the inverse Darcy parameter D -. It is interesting to note that lesser the permeability of the porous medium lower the magnitude of the resultant velocity. i.e., the resultant velocity reduces throughout the fluid region with increase in the inverse Darcy parameter D -. Both the velocity components u and w enhances with increase in the relaxation time entire fluid region. These displayed in the figures ( and. The resultant velocity enhances throughout the fluid region with increase in the relaxation time. The shear stresses on the upper and lower plates have been calculated with reference to variations in the governing parameters and are tabulated in the tables (I-IV. On the upper plate the magnitude of the stresses with increase in M and magnitude of the stresses y x enhances, while it reduces with increase in the inverse Darcy parameter D -. The enhances with increase in for all governing parameters M, D - and (tables. I-II. On the lower plate the magnitude of the stresses and IV. x and y enhances with increase in M, while these reduces with increase in the inverse Darcy parameter D - (tables. III-IV. CONCLUSIONS. The resultant velocity reduces throughout the fluid region with increase in the intensity of the magnetic field (the Hartmann number M.. Lesser the permeability of the porous medium lower the magnitude of the resultant velocity. i.e., the resultant velocity reduces throughout the fluid region with increase in the inverse Darcy parameter D -.. Both the velocity components u and w and the resultant velocity enhances with increase in the relaxation time in the entire fluid region.. On the upper plate the magnitude of the stresses x enhances with increase in M and, while it reduces with increase in the inverse Darcy parameter D -.. The magnitude of the stresses enhances with increase in for all governing parameters M, D - y and. On the lower plate the magnitude of the stresses x and enhances with increase in y M, and, while these reduces with increase in the inverse Darcy parameter D -. Vol., Issue, pp. -

7 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 V. GRAHS AND TABLES.. u M= M= M= M= Fig. : The velocity profile for u with M., D =, u D ¹= D ¹= D ¹= D ¹=....8 Fig. : The velocity profile for u with D -., M=... u.. β= β= β= β= Fig. : The velocity profile for u with. D =, M= Vol., Issue, pp. -

8 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: v M= M= M=8 M= -. Fig. : The velocity profile for w with M., D =, v D ¹= D ¹= D ¹= D ¹= -. v Fig. : The velocity profile for w with D -., M=....8 β= β= β= β= Fig. : The velocity profile for w with. D =, M= Vol., Issue, pp. -

9 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 Table I: The shear stresses ( x on the upper plate = I II III IV V VI VII I II III IV V VI VII M 8 D 8 Table II: The shear stresses ( y on the upper plate = I II III IV V VI VII I II III IV V VI VII M 8 D 8 Table III: The shear stresses ( x on the lower plate = I II III IV V VI VII I II III IV V VI VII M 8 D 8 Table IV: The shear stresses ( y on the lower plate = I II III IV V VI VII I II III IV V VI VII M 8 D 8 Vol., Issue, pp. -

10 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 ACKNOWLEDGEMENTS The authors very much thankful to authorities of JNTU, Anantapur, Andhra radesh, India, providing necessary facilities to have done this work and IJAET journal for the support to develop this document. REFERENCES [].. uri, Rotary flowof an elastico-viscous fluid on an oscillating plate, Journal of Applied Mathematics and Mechanics, Vol., No., pp., 9. []. M. Hussain, T. Hayat, S. Asghar, and C. Fetecau, Oscillatory flows of second grade fluid in a porous space, Nonlinear Analysis: Real World Applications, Vol., No., pp.,. []. C. Fetecau, S. C. rasad, and K. R. Rajagopal, A note on the flow induced by a constantly accelerating plate in an Oldroyd-B fluid, Applied Mathematical Modelling, Vol., No., pp.,. []. W. Tan and T. Masuoka, Stokes first problem for a second grade fluid in a porous half-space with heated boundary, International Journal of Non-Linear Mechanics, Vol., No., pp.,. []. C. Fetecau, M. Athar, and C. Fetecau, Unsteady flow of a generalied Maxwell fluid with fractional derivative due to a constantly accelerating plate, Computers andmathematics with Applications, Vol., No., pp. 9, 9. []. M. Husain, T. Hayat, C. Fetecau, and S. Asghar, On accelerated flows of an Oldroyd B fluid in a porous medium, Nonlinear Analysis: Real World Applications, Vol. 9, No., pp. 9 8, 8. []. M. Khan, E. Naheed, C. Fetecau, and T. Hayat, Exact solutions of starting flows for second grade fluid in a porous medium, International Journal of Non-LinearMechanics, Vol., No. 9, pp , 8. [8]. F. Salah, Z. A. Ai, and D. L. C. Ching, New exact solution for Rayleigh-Stokes problem ofmaxwell fluid in a porous medium and rotating frame, Results in hysics, Vol., No.,pp. 9,. [9]. M. Khan, M. Saleem, C. Fetecau, and T. Hayat, Transient oscillatory and constantly accelerated non- Newtonian flow in a porous medium, International Journal of Non-Linear Mechanics, Vol., No., pp. 9,. []. F. Salah, Z.Abdul Ai, andd. L.C.Ching, New exact solutions for MHD transient rotating flow of a second-grade fluid in a porous medium, Journal of Applied Mathematics, Vol., Article ID 8, 8 pages,. []. C. Fetecau, T. Hayat, M. Khan, and C. Fetecau, Erratum: Unsteady flow of an Oldroyd-B fluid induced by the impulsive motion of a plate between two side walls perpendicular to the plate, Acta Mechanica, Vol., No., pp. 9,. []. C. Fetecau, T. Hayat, J. Zierep, and M. Sajid, Energetic balance for the Rayleigh-Stokes problem of an Oldroyd-B fluid, Nonlinear Analysis: Real World Applications, Vol., No., pp.,. []. K. R. Rajagopal and A. S.Gupta, On a class of exact solutions to the equations of motion of a second grade fluid, International Journal of Engineering Science, Vol. 9, No., pp. 9, 98. []. M. E. Erdoˇgan and C. E. Imrak, On unsteady unidirectional flows of a second grade fluid, International Journal of Non-Linear Mechanics, Vol., No., pp. 8,. []. F. Salah, Z. A. Ai, and D. L. C. Ching, Accelerated flows of a magnetohydrodynamic (MHD second grade fluid over an oscillating plate in a porous medium and rotating frame, International Journal of hysical Sciences, Vol., No., pp. 8 8,. []. C. Fetecau andc. Fetecau, Starting solutions for some unsteady unidirectional flows of a second grade fluid, International Journal of Engineering Science, Vol., No., pp. 8 89,. []. T. Hayat, K. Hutter, S. Asghar, and A.M. Siddiqui, MHD flows of an Oldroyd-B fluid, Mathematical and Computer Modelling, Vol., No. 9-, pp ,. [8]. T.Hayat, S.Nadeem, S.Asghar, anda.m. Siddiqui, Fluctuating flow of a third-grade fluid on a porous plate in a rotating medium, International Journal of Non-Linear Mechanics, Vol., No., pp. 9 9,. [9]. S. Abelman, E. Momoniat, and T. Hayat, Steady MHD flow of a third grade fluid in a rotating frame and porous space, Nonlinear Analysis: Real World Applications, Vol., No., pp. 8, 9. []. Faisal Salah, Zainal Abdul Ai, Mahad Ayem, and Dennis Ling Chuan Ching, MHD Accelerated Flow of Maxwell Fluid in a orous Medium and Rotating Frame, ISRN Mathematical hysics, Vol., Article ID 88, pages,, []. Hayat.T, C.Fetecau, M.Sajid., hysics Letters A, Vol., pp. 9-, 8. []. M.VeeraKrishna, S.V.Suneetha and R.Sivarasad, Hall current effects on unsteady MHD flow of rotating Maxwell fluid through a porous medium, Ultra Scientist of hysical Sciences, Vol. (M, pp. -,. Vol., Issue, pp. -

11 International Journal of Advances in Engineering & Technology, July,. IJAET ISSN: 9 Appendix s b b M b β s D β s φ, s β β ω i ω iβ ω s β (M β ( n π,b ( n π, s, b, s D β s β φ, b β s s β s β, s, b, b β s ( n π β β β ( n π βω i ω iβ ω β ω i ω iβ ω β,,, AUTHORS BIOGRAHY L. Sreekala, presently working as a Asst rofessor in the Department of Mathematics in Chiranjeevi Reddy Institute of Technology, Anantapur, Andhra radesh India, I have seven years of experience in teaching and three years in Research. I am doing my h.d in area of fluid dynamics. I have published many papers in national and international well reputed journals. M. Veera Krishna received the B.Sc. degree in Mathematics, hysics and Chemistry from the Sri Krishnadevaraya University, Anantapur, Andhra radesh, India in 998, the M.Sc. in Mathematics in, the M.hil and h.d. degree in Mathematics from same, in and 8, respectively. Currently, He is an in-charge of Department of Mathematics at Rayalaseema University, Kurnool, Andhra radesh, India. His teaching and research areas include Fluid mechanics, Heat transfer, MHD flows and Data mining techniques. He awarded h.d from Monad University, Hapur (U., India and 8 M.hils from DDE, S.V. University, Tirupati, A.., He has published research papers in national and international well reputed journals. He has presented 8 papers in National and International seminars and conferences. He attended four national level workshops. He is a life member of Indian Society of Theoretical and Applied Mechanics (ISTAM. L. Hari Krishna acquired M.Sc. degree in mathematics from S.V. University, Tirupati in the year 998. He obtained his M.hil from M.K. University, Madhurai in the year. He obtained his h.d in the area of Fluid Dynamics from JNTU, Anantapur in. He has got years of experience in teaching to engineering students and seven years of research experience. He has published 8 international publications. He has presented 8 papers in National and International seminars. He has attend two national level workshops. He was a Editorial and Article review board member for AES Journal in Engineering Technology and Sciences (-. He has periyar university Guide ship approval and successfully guided students to take their M.hils. He has memberships in professional bodies. K. Keshava Reddy, presently working as professor of Mathematics in JNT University college of Engineering Anantapur, He has years of experience in teaching and years in research, He obtained his h.d degree in mathematics from prestigious University Banaras Hindu University varanasi, His areas of interest include functional Anaysis. Optimiation Techniques, Data mining, Neural Networks and Fuy logic. He produced h.d, M.hil and has published more than Research papers in National and International Journals and conference. He authored books on engineering mathematics and Mathematical Methods for various Mathematics for JNTUA both at UG level and G level presently he is the chairman, G Board of studies for Mathematics of JNTUA. He is a member of Board of studies for Mathematics of various universities in India. Vol., Issue, pp. -